The most usual case of a crowded field is a star cluster. The accuracy of photometry in such a situation as been assessed by several authors for HST images (see e.g. Malumuth et al. 1991 or Sodemann & Thomsen 1997). These studies showed that the errors on the magnitude of fainter stars could become large very quickly. For instance, Sodemann & Thomsen (1997) studied two different crowded field observed with the HST and showed that the magnitudes of most stars were overestimated and that the errors could easily reach several tenths of a magnitude for faint stars. In this section, we assess the accuracy of photometry in crowded fields in the case of adaptive optics observations. This case is linked to the previous one, but in a way that can be complex.
We created several images of random crowded fields with a uniform mean density using the IRAF/ARTDATA package. As point spread functions, we used observed images of HD5980 taken in the H and K bands. The Strehl ratios were respectively 0.13 and 0.32 and the signal to noise ratios 5500 and 7000. The integration time was 200 seconds in both cases and the pixel size 0.050 arcsec. We used the Bahcall and Soneira luminosity function provided by the package with a 10-magnitude range (Bahcall & Soneira 1980). This function with its default parameters gives a good fit to the observed main sequence in several nearby globular cluster. To simulate different densities of stars, we changed either the total number of stars or the size of the field of view, but kept a fixed spatial scale. We were careful to keep a number of stars large enough (at least 50) to provide a good representation of the average behaviour.
After creation of an image, we used the IRAF/NOAO DAOPHOT package to compute the magnitude of its stars. This was done with two different point spread functions. First, the right one, which enabled us to assess the performances of the DAOPHOT algorithm itself and the problems due to the very complex shape of the PSF, especially the presence of numerous residual features around the core, but also the unavoidable numerical errors appearing during the fitting process. Then, a different PSF was used to study the problems due to the mismatch between the actual PSF and the one obtained from a calibration star (required when no suitable PSF can be found in the cluster). As a second PSF, we used an image of the reference star of HD5980 in the same band. The integration times for these reference stars were respectively 120 and 100 seconds, the Strehl ratios 0.35 and 0.15, and the signal to noise ratios 45000 and 35000. The time delay between images was 15 minutes in H and 20 minutes in K. DAOPHOT yielded magnitudes that could be compared to the real ones used in the simulation of the star field. We worked out the photometric errors for 7 different densities: 0.15, 0.5, 0.75, 1.5, 3, 5 and 15 stars per arcsec2 (it would, of course, be impossible to do photometry on uncompensated images for the higher densities). No noise was added to the artificial image as we were concentrating on the problems due to the shape of the PSF, not on the influence of other sources of noise. Note that we supposed the positions of the stars known before application of DAOPHOT as a starting point, which would for example be the case if a star detection algorithm using a deconvolved image had been applied first (see e.g. Snell 1996). We nevertheless let the algorithm try to recentre the stars afterwards.
Figures 8 (click here) and 9 (click here) present typical results obtained through this procedure. The magnitude estimated by DAOPHOT for each star is given as a function of its real magnitude, and this is compared to the theoretical line that would be obtained if the measurement were perfectly accurate. Figure 8 has been obtained with an H-band PSF and two different stellar densities: 1.5 and 15 stars per arcsec2. DAOPHOT was provided with the right PSF, the same as used to create the artificial cluster (an image of HD5980). Figure 9 (click here) has been obtained with K-band PSFs for a density of 15 stars per arcsec2. In this case, two results are compared, the ones obtained when using the right PSF and the ones obtained when using a wrong PSF (the reference star SAO255763 instead of HD5980).
Figure 8: Stellar magnitudes estimated by DAOPHOT in a star cluster as a function
of the real magnitudes. Two densities are presented:
1.5 stars per arcsec2 and 15 stars per arcsec2.
The dotted line corresponds to a perfect agreement between estimated and
real magnitudes and the zero point for the magnitude scale is arbitrary.
Theses results have been obtained by providing DAOPHOT with the right
PSF, the same used to create the artificial cluster.
The PSF was an H-band image of HD5980
Figure 9: Stellar magnitudes estimated by DAOPHOT in a star cluster as a function
of the real magnitudes. Two cases are presented: one where DAOPHOT
was provided with the right PSF (a K-band image of HD5980)
and one when the algorithm used a wrong PSF (a K-band image of SAO255763).
The density was 15 stars per arcsec2.
The dotted line corresponds to a perfect agreement between estimated and
real magnitudes and the zero point for the magnitude scale is arbitrary
Consider first the case where DAOPHOT is provided with the right PSF, as illustrated in Fig. 8 (click here). The first thing to notice is the fact that the magnitudes of the brightest stars are in general accurately estimated, even if a few of them are clearly underestimated. There is nevertheless a scatter that can be considered large for many astronomical purposes. We computed the errors on the magnitudes of the brightest stars for each density and PSF. The rms error in the magnitude determination varied between 0.05 and 0.1 magnitude for densities lower than a few stars per arcsec2 and could reach 0.15 for higher densities. This was true for both H and K band PSFs. From this, we can already conclude that crowded fields do not allow a very good photometric precision. Even for the brightest stars, the accuracy is at best 0.05 magnitudes, which can already be a major drawback for a large number of projects.
The second feature visible in Fig. 8 (click here) is the fact that the errors increase as the star become fainter, and as a consequence the faintest stars can have their flux overestimated by several magnitudes. Of course, the lower the density, the fainter the level at which this increase of the errors occurs. Not all the faint stars are affected and the curves present a large scatter. This could be expected as the magnitude of a star is not the only important parameter in the error on its photometric estimation. Other factors are the distances to the nearest bright stars and the magnitudes of these objects.
This behaviour where stars, especially the faintest ones, have their intensity systematically overestimated can be attributed to three effects, two of which are always present in crowded fields and are not due to the use of adaptive optics. The first effect is confusion: two stars that are too close cannot be distinguished and are therefore considered by DAOPHOT as a single object, which directly leads to an overestimation of the flux. The second effect is blending: stars too faint to be detected can increase the estimated brightness of the brighter sources, which also lead to an overestimation. These two effects are not linked to the use of adaptive optics. On the contrary, the sharper the PSF, the less confusion there is, and the more faint sources can be detected. For this reason, adaptive optics yields a large improvement compared to normal ground-based images (see the end of this section for an illustration).
The source of uncertainty linked to the use of adaptive optics is the presence of numerous residual features in the PSF. As a consequence, some of the light contained in the residual features associated with the bright stars of the field might be redistributed into the faint stars, thus increasing their measured brightness and at the same time slightly decreasing the measured flux of the bright stars. This would be even more true when a wrong PSF is used by DAOPHOT, because of the fluctuations of these features. The importance of this effect can be estimated from Fig. 9 (click here), which compares the results in given conditions for an estimation with the right PSF and one with the wrong PSF. It can be seen that more bright sources are underestimated when DAOPHOT uses the wrong PSF, and the rms error for these sources is therefore larger. The difference is less obvious as far as fainter sources are concerned. The increase in the estimated error is then similar and no obvious difference is visible (which will be confirmed below in this section). The residual features therefore introduce some uncertainty for the brightest stars, but confusion and blending are still the major sources of noise for fainter sources. Note that the presence of large errors in the estimated magnitude of the brightest stars only occured for the highest densities, say a few stars par arcsec2.
In order to study the behaviour of photometry in crowded fields as a function of the stellar densities, we worked out for each density, the difference in magnitude relative to the brightest star at which a given level of error was reached. Given that the error are already quite significant for the brightest stars (between 0.05 and 0.15 magnitude), we chose a value of 0.2 magnitude as a threshold for the error. The measurement of the limiting difference in magnitude was carried out by viewing the previous curves with an adequate magnification. Given the very irregular distribution of points as illustrated in Fig. 8, this limiting difference is sometimes difficult to estimate precisely, and the uncertainty in it is of the order of 0.5 magnitude or a little more.
Figure 10: The difference in magnitude relative to the brightest star at
which the error on the estimated flux reaches a level of 0.2 magnitude,
as a function of the stellar density. Four sets of values are shown,
corresponding to PSFs obtained in the H and K band and to
either the right or the wrong PSF for the application of DAOPHOT.
The uncertainty on each point is of the order of 0.5 magnitude
Figure 10 (click here) gives the limiting magnitude defined above as a function of the stellar density. Four curves are presented, two with H band PSFs, two with K band PSFs, two when DAOPHOT used the right PSF and two when the algorithm used a wrong PSF. It can broadly be seen that for low densities, an error of 0.2 is reached for stars between 8 and 9 magnitudes fainter than the brightest source in the field. Note that the limiting magnitudes in this case is partially due to the limited signal to noise of the PSFs used for the creation of the artificial cluster (5500 in H and 7000 in K, corresponding to 9.4 and 9.6 magnitudes between the peak and the background noise). When the density increases, the limiting difference in magnitude goes down, reaching about 3 in H and 6 in K for a density of a few stars per arcsec2. At more than 10 stars per arcsec2, the limiting difference reaches 2 in H and about 4 in K. But note that in the case of mismatched PSFs in the K band, an error of 0.2 magnitude is already present for the brightest sources. This might be due to a high level of fluctuations in the residual features as explained before. Several other features can be noted. First, with the exception of the highest density, the curves obtained in a given band with the right or the wrong PSF are very similar. This confirms that the mismatch between PSF is less important than the confusion and blending problems at this level. Second, still with the exception of the highest density, the results are better in the K band. This is clearly the result of a better Strehl ratio, which decreases the effects of confusion and blending.
As a conclusion, we can note that photometry in a crowded field
leads to inaccurate results, with errors between 0.05 and 0.1 magnitude
for the brightest stars, and reaching several tenths of a magnitude and
even one magnitude for fainter stars. But adaptive optics
is not directly responsible for this problem. On the contrary, the
use of this technique strongly reduces confusion
and blending and therefore allows
a linear behaviour until fainter sources. To compare adaptive optics
and normal results, we also created artificial images of a cluster using
a seeing-limited PSF with a gaussian shape and a FWHM of
1.0 arcsec
We chose two densities, 1 and 10 stars per arcsec2, and
performed the same procedure as for the adaptive optics images, using
the same PSF for the fitting algorithm and the creation of the artificial
image. Figure 10 (click here)
shows that adaptive optics images yield a precision better than
0.2 magnitude down to 
for a density of
one star per arcsec2 and down to 
or 4.5 for
10 stars per arcsec2 (assuming the right PSF is used).
Moreover, for the brightest stars, the errors in the estimated
magnitudes are respectively 0.05 and 0.1 magnitude or so.
For the seeing limited images, the measurements of the brightest stars
were already wrong by about 0.5 magnitude for both densities.
In the case of a density of one star per arcsec2,
for stars fainter than 
, some measurements
were affected by errors between 1 and 2 magnitudes, and beyond
, the errors kept on increasing rapidly. For the higher
density, an error of 1 magnitude already occured at .
The use of adaptive optics therefore brings a significant improvement
in the photometry of crowded fields compared to seeing-limited image,
but it has to be kept in mind that the error on the flux of faint
sources can rapidly become very large.
We also assessed the performances of point spread function fitting in clusters as far as angular anisoplanatism is concerned. We used the same simulated PSFs as in the study of angular anisoplanatism in aperture photometry. For each value of D/r0 and each separation, we tried to fit the on-axis PSF to the off-axis one using DAOPHOT. Comparing the result with a fit of the on-axis PSF to itself gave us an estimation of the error due to PSF distortions induced by angular anisoplanatism. Note that DAOPHOT needed an aperture size to make a first guess of the magnitude. We checked that the final estimate was independent of the choice of this aperture size.
| Band | 5'' | 10'' | 15'' | 20'' |
| J | 0.07 | 0.28 | 0.53 | 0.95 |
| H | 0.05 | 0.17 | 0.34 | 0.62 |
| K | 0.02 | 0.09 | 0.17 | 0.32 |
|
|
Table 5 (click here) presents the results of this study. The error in the magnitude estimation is given in the J, H and K bands for four different distances to the centre. Comparing Table 5 to Table 4 (click here) shows that the level of error is much higher in this case, by about a factor 2 compared to the error for aperture photometry using an aperture size defined by the first dark ring. This high level is obviously a important drawback. Even in K for a distance of 5 arcsec, the error is already about 0.02 magnitude. Note that some work is in progress to enable DAOPHOT to take into account large variations of the PSF in the field of view and the situation should therefore improve (Stetson 1994). Also note in a real case, the level of variation of the PSF depends on several factors like the seeing, the number and strengths of turbulent layers or the central obscuration of the telescope. The results obtained in Table 5 should therefore only be considered as illustrative.